Open problems in PDE models for knowledge-based animal movement via nonlocal perception and cognitive mapping

The inclusion of cognitive processes, such as perception, learning and memory, are inevitable in mechanistic animal movement modelling. Cognition is the unique feature that distinguishes animal movement from mere particle movement in chemistry or physics. Hence, it is essential to incorporate such knowledge-based processes into animal movement models. Here, we summarize popular deterministic mathematical models derived from first principles that begin to incorporate such influences on movement behaviour mechanisms. Most generally, these models take the form of nonlocal reaction-diffusion-advection equations, where the nonlocality may appear in the spatial domain, the temporal domain, or both. Mathematical rules of thumb are provided to judge the model rationality, to aid in model development or interpretation, and to streamline an understanding of the range of difficulty in possible model conceptions. To emphasize the importance of biological conclusions drawn from these models, we briefly present available mathematical techniques and introduce some existing "measures of success" to compare and contrast the possible predictions and outcomes. Throughout the review, we propose a large number of open problems relevant to this relatively new area, ranging from precise technical mathematical challenges, to more broad conceptual challenges at the cross-section between mathematics and ecology. This review paper is expected to act as a synthesis of existing efforts while also pushing the boundaries of current modelling perspectives to better understand the influence of cognitive movement mechanisms on movement behaviours and space use outcomes.

Spatial memory predicts home range size and predation risk in pheasants

Most animals confine their activities to a discrete home range, long assumed to reflect the fitness benefits of obtaining spatial knowledge about the landscape. However, few empirical studies have linked spatial memory to home range development or determined how selection operates on spatial memory via the latter's role in mediating space use. We assayed the cognitive ability of juvenile pheasants (Phasianus colchicus) reared under identical conditions before releasing them into the wild. Then, we used high-throughput tracking to record their movements as they developed their home ranges, and determined the location, timing and cause of mortality events. Individuals with greater spatial reference memory developed larger home ranges. Mortality risk from predators was highest at the periphery of an individual's home range in areas where they had less experience and opportunity to obtain spatial information. Predation risk was lower in individuals with greater spatial memory and larger core home ranges, suggesting selection may operate on spatial memory by increasing the ability to learn about predation risk across the landscape. Our results reveal that spatial memory, determined from abstract cognitive assays, shapes home range development and variation, and suggests predation risk selects for spatial memory via experience-dependent spatial variation in mortality.

Reinforced diffusions as models of memory-mediated animal movement

How memory shapes animals' movement paths is a topic of growing interest in ecology, with connections to planning for conservation and climate change. Empirical studies suggest that memory has both temporal and spatial components, and can include both attractive and aversive elements. Here, we introduce reinforced diffusions (the continuous time counterpart of reinforced random walks) as a modelling framework for understanding the role that memory plays in determining animal movements. This framework includes reinforcement via functions of time before present and of distance away from a current location. Focusing on the interplay between memory and central place attraction (a component of home ranging behaviour), we explore patterns of space usage that result from the reinforced diffusion. Our efforts identify three qualitatively different behaviours: bounded wandering behaviour that does not collapse spatially, collapse to a very small area, and, most intriguingly, convergence to a cycle. Subsequent applications show how reinforced diffusion can create movement trajectories emulating the learning of movement routes by homing pigeons and consolidation of ant travel paths. The mathematically explicit manner with which assumptions about the structure of memory can be stated and subsequently explored provides linkages to biological concepts like an animal's 'immediate surroundings' and memory decay.

What’s in a resource gradient? Comparing alternative cues for foraging in dynamic environments via movement, perception, and memory

Consumers must track and acquire resources in complex landscapes. Much discussion has focused on the concept of a ‘resource gradient’ and the mechanisms by which consumers can take advantage of such gradients as they navigate their landscapes in search of resources. However, the concept of tracking resource gradients means different things in different contexts. Here, we take a synthetic approach and consider six different definitions of what it means to search for resources based on density or gradients in density. These include scenarios where consumers change their movement behavior based on the density of conspecifics, on the density of resources, and on spatial or temporal gradients in resources. We also consider scenarios involving non-local perception and a form of memory. Using a continuous space, continuous time model that allows consumers to switch between resource-tracking and random motion, we investigate the relative performance of these six different strategies. Consumers’ success in matching the spatiotemporal distributions of their resources differs starkly across the six scenarios. Movement strategies based on perception and response to temporal (rather than spatial) resource gradients afforded consumers with the best opportunities to match resource distributions. All scenarios would allow for optimization of resource-matching in terms of the underlying parameters, providing opportunities for evolutionary adaptation, and links back to classical studies of foraging ecology.

Time-dependent memory and individual variation in Arctic brown bears (Ursus arctos)

BACKGROUND

Animal movement modelling provides unique insight about how animals perceive their landscape and how this perception may influence space use. When coupled with data describing an animal's environment, ecologists can fit statistical models to location data to describe how spatial memory informs movement.

METHODS

We performed such an analysis on a population of brown bears (Ursus arctos) in the Canadian Arctic using a model incorporating time-dependent spatial memory patterns. Brown bear populations in the Arctic lie on the periphery of the species' range, and as a result endure harsh environmental conditions. In this kind of environment, effective use of memory to inform movement strategies could spell the difference between survival and mortality.

RESULTS

The model we fit tests four alternate hypotheses (some incorporating memory; some not) against each other, and we found a high degree of individual variation in how brown bears used memory. We found that 71% (15 of 21) of the bears used complex, time-dependent spatial memory to inform their movement decisions.

CONCLUSIONS

These results, coupled with existing knowledge on individual variation in the population, highlight the diversity of foraging strategies for Arctic brown bears while also displaying the inference that can be drawn from this innovative movement model.

Spatial Memory Drives Foraging Strategies of Wolves, but in Highly Individual Ways

The ability of wild animals to navigate and survive in complex and dynamic environments depends on their ability to store relevant information and place it in a spatial context. Despite the centrality of spatial memory, and given our increasing ability to observe animal movements in the wild, it is perhaps surprising how difficult it is to demonstrate spatial memory empirically. We present a cognitive analysis of movements of several wolves (Canis lupus) in Finland during a summer period of intensive hunting and den-centered pup-rearing. We tracked several wolves in the field by visiting nearly all GPS locations outside the den, allowing us to identify the species, location and timing of nearly all prey killed. We then developed a model that assigns a spatially explicit value based on memory of predation success and territorial marking. The framework allows for estimation of multiple cognitive parameters, including temporal and spatial scales of memory. For most wolves, fitted memory-based models outperformed null models by 20 to 50% at predicting locations where wolves chose to forage. However, there was a high amount of individual variability among wolves in strength and even direction of responses to experiences. Some wolves tended to return to locations with recent predation success—following a strategy of foraging site fidelity—while others appeared to prefer a site switching strategy. These differences are possibly explained by variability in pack sizes, numbers of pups, and features of the territories. Our analysis points toward concrete strategies for incorporating spatial memory in the study of animal movements while providing nuanced insights into the behavioral strategies of individual predators.

Prey encounters and spatial memory influence use of foraging patches in a marine central place forager

Given the patchiness and long-term predictability of marine resources, memory of high-quality foraging grounds is expected to provide fitness advantages for central place foragers. However, it remains challenging to characterize how marine predators integrate memory with recent prey encounters to adjust fine-scale movement and use of foraging patches. Here, we used two months of movement data from harbour seals (Phoca vitulina) to quantify the repeatability in foraging patches as a proxy for memory. We then integrated these data into analyses of fine-scale movement and underwater behaviour to test how both spatial memory and prey encounter rates influenced the seals' area-restricted search (ARS) behaviour. Specifically, we used one month's GPS data from 29 individuals to build spatial memory maps of searched areas and archived accelerometery data from a subset of five individuals to detect prey catch attempts, a proxy for prey encounters. Individuals were highly consistent in the areas they visited over two consecutive months. Hidden Markov models showed that both spatial memory and prey encounters increased the probability of seals initiating ARS. These results provide evidence that predators use memory to adjust their fine-scale movement, and this ability should be accounted for in movement models.

Memory drives the formation of animal home ranges: Evidence from a reintroduction

Most animals live in home ranges, and memory is thought to be an important process in their formation. However, a general memory-based model for characterising and predicting home range emergence has been lacking. Here, we use a mechanistic movement model to: (1) quantify the role of memory in the movements of a large mammal reintroduced into a novel environment, and (2) predict observed patterns of home range emergence in this experimental setting. We show that an interplay between memory and resource preferences is the primary process influencing the movements of reintroduced roe deer (Capreolus capreolus). Our memory-based model fitted with empirical data successfully predicts the formation of home ranges, as well as emergent properties of movement and spatial revisitation observed in the reintroduced animals. These results provide a mechanistic framework for combining memory-based movements, resource preferences, and the formation of home ranges in nature.

An innovative approach to identify environmental variables with conservation priorities in habitat patches

Conservation of habitat patches and the related environment benefits both the focal species and human well-being. Many indices use the dispersal range to identify habitat patches with conservation priorities. However, there lacks approaches to identify environmental variables with conservation priorities (noted as target variables) in those identified patches. Therefore, this paper proposes an approach to identify environmental variables with conservation priorities in habitat patches using perception range and introduces the related assumption. It is assumed the agents select habitats based on their prior preference and perceived information in their perception ranges, which avoids the omniscient assumption of agents. Based on such assumptions, the proposed approach identifies the target variables by approximating how animals identify their habitats. It highlights the use of perception range and identifies target variables using the maximum information gain. The variables that contribute the largest reduction of uncertainty are regarded as the target variables in the habitat patches. Taking the Common Moorhen (Gallinula chloropus) living in Tianjin, China as the case, different scenarios with 100 m, 250 m and 500 m perception ranges are designed to illustrate the feasibility of the proposed approach. The proposed approach identifies the normalized vegetation index, rather than the distance to water surface, is the target variable in 42.3%, 58.9% and 72.1% habitat patches with given perception ranges. Adjustments are made on areas within the given perception range of each patch. More grid cells that has increased suitability index can be found in scenarios given 250 m perception range, which indicates the conservation area is not always the large the better. Optimizations are expected on both a better approximation method and a more thorough hypothesis of using perception range.

Hierarchical, Memory-Based Movement Models for Translocated Elk (Cervus canadensis)

The use of spatial memory is well-documented in many animal species and has been shown to be critical for the emergence of spatial learning. Adaptive behaviors based on learning can emerge thanks to an interdependence between the acquisition of information over time and movement decisions. The study of how spatio-ecological knowledge is constructed throughout the life of an individual has not been carried out in a quantitative and comprehensive way, hindered by the lack of knowledge of the information an animal already has of its environment at the time monitoring begins. Identifying how animals use memory to make beneficial decisions is fundamental to developing a general theory of animal movement and space use. Here we propose several mobility models based on memory and perform hierarchical Bayesian inference on 11-month trajectories of 21 elk after they were released in a completely new environment. Almost all the observed animals exhibited preferential returns to previously visited patches, such that memory and random exploration phases occurred. Memory decay was mild or negligible over the study period. The fact that individual elk rapidly become used to a relatively small number of patches was consistent with the hypothesis that they seek places with predictable resources and reduced mortality risks such as predation.

Learning and Animal Movement

Integrating diverse concepts from animal behavior, movement ecology, and machine learning, we develop an overview of the ecology of learning and animal movement. Learning-based movement is clearly relevant to ecological problems, but the subject is rooted firmly in psychology, including a distinct terminology. We contrast this psychological origin of learning with the task-oriented perspective on learning that has emerged from the field of machine learning. We review conceptual frameworks that characterize the role of learning in movement, discuss emerging trends, and summarize recent developments in the analysis of movement data. We also discuss the relative advantages of different modeling approaches for exploring the learning-movement interface. We explore in depth how individual and social modalities of learning can matter to the ecology of animal movement, and highlight how diverse kinds of field studies, ranging from translocation efforts to manipulative experiments, can provide critical insight into the learning process in animal movement.

Knowing your neighbours: How memory-mediated conspecific avoidance influences home ranges

In Focus: Ellison, N., Hatchwell, B. J., Biddiscombe, S. J., Napper, C. J., & Potts, J. R. (2020). Mechanistic home range analysis reveals drivers of space use patterns for a non-territorial passerine. Journal of Animal Ecology. https://doi.org/10.1111/1365-2656.13292. Most animals for which space use has been studied restrict their movements into a constrained spatial area: their home range. The ubiquity of this space-use pattern suggests that home ranges are adaptive in a wide range of ecological contexts, and that they likely arise from general biological mechanisms. In this issue, Ellison et al. use a mechanistic home range analysis (MHRA) to uncover the drivers underlying home range patterns in a passerine that is non-territorial. They show that a model integrating both resource preferences (specifically, an attraction to woodland centre), and memory-mediated conspecific avoidance can capture the space-use patterns observed in a wild population of long-tailed tits Aegithalos caudatus. In doing so, their analysis extends the applicability of MHRA to capturing and predicting home range patterns beyond the previously studied cases where spatially exclusive home ranges emerge from scent mark-mediated avoidance responses to neighbouring groups.

Spatial movement with distributed memory

Diffusion has been widely applied to model animal movement that follows Brownian motion. However, animals typically move in non-Brownian ways due to their perceptual judgment. Spatial memory and cognition recently have received much attention in characterizing complicated animal movement behaviours. Explicit spatial memory is modeled via a distributed delayed diffusion term in this paper. The distributed time represents the memory growth and decay over time, and the spatial nonlocality reflects the dependence of spatial memory on location. When the temporal delay kernel is weak under the assumption that animals can immediately acquire knowledge and memory decays over time, the equation is equivalent to a Keller-Segel chemotaxis model. For the strong kernel with learning and memory decay stages, rich spatiotemporal dynamics, such as Turing and checker-board patterns, appear via spatially non-homogeneous steady-state and Hopf bifurcations.

Memory-driven movement model for periodic migrations

We propose a model for memory-based movement of an individual. The dynamics are modeled by a stochastic differential equation, coupled with an eikonal equation, whose potential depends on the individual's memory and perception. Under a simple periodic environment, we discover that both long and short-term memory with appropriate time scales are essential for producing expected periodic migrations.

Crossover in spreading behavior due to memory in population dynamics

The reaction-diffusion equation is one of the possible ways for modeling animal movement, where the reactive part stands for the population growth and the diffusive part for random dispersal of the population. However, a reaction-diffusion model may not represent all aspects of the spatial dynamics, because of the existence of distinct mechanisms that can affect the movement, such as spatial memory, which results in a bias for one direction of dispersal. This bias is modeled through an advective term on an advection-reaction-diffusion equation. Thus, considering the effects of memory on the population spread, we propose a model composed of a coupled partial differential equation system with two equations: one for the population dynamics and the other for the memory density distribution. For the population growth, we use either the exponential or logistic growth function. The analytic approach shows that for the exponential and logistic growth, the minimum traveling wave speeds are the same with or without memory dynamics in which the variation of memory is infinitesimal. From the numerical analysis, we explore how our parameters, memory, growth rate, and carrying capacity, affect the population redistribution. The combinations of these parameters result in a redistribution pattern of the population associated with either diffusive or superdiffusive and imply the dispersal is faster than the diffusion. Further, in the parameter-space defined by memory and growth rate, we have shown that memory is a factor that switches the dynamics between two spreading behaviors, one faster than the other.

Territory surveillance and prey management: Wolves keep track of space and time

Identifying behavioral mechanisms that underlie observed movement patterns is difficult when animals employ sophisticated cognitive‐based strategies. Such strategies may arise when timing of return visits is important, for instance to allow for resource renewal or territorial patrolling. We fitted spatially explicit random‐walk models to GPS movement data of six wolves (Canis lupus; Linnaeus, 1758) from Alberta, Canada to investigate the importance of the following: (1) territorial surveillance likely related to renewal of scent marks along territorial edges, to reduce intraspecific risk among packs, and (2) delay in return to recently hunted areas, which may be related to anti‐predator responses of prey under varying prey densities. The movement models incorporated the spatiotemporal variable “time since last visit,” which acts as a wolf's memory index of its travel history and is integrated into the movement decision along with its position in relation to territory boundaries and information on local prey densities. We used a model selection framework to test hypotheses about the combined importance of these variables in wolf movement strategies. Time‐dependent movement for territory surveillance was supported by all wolf movement tracks. Wolves generally avoided territory edges, but this avoidance was reduced as time since last visit increased. Time‐dependent prey management was weak except in one wolf. This wolf selected locations with longer time since last visit and lower prey density, which led to a longer delay in revisiting high prey density sites. Our study shows that we can use spatially explicit random walks to identify behavioral strategies that merge environmental information and explicit spatiotemporal information on past movements (i.e., “when” and “where”) to make movement decisions. The approach allows us to better understand cognition‐based movement in relation to dynamic environments and resources.

Emergent weak home-range behaviour without spatial memory

Space-use problems have been well investigated. Spatial memory capacity is assumed in many home-range algorithms; however, actual living things do not always exploit spatial memory, and living entities can exhibit adaptive and flexible behaviour using simple cognitive capacity. We have developed an agent-based model wherein the agent uses only detected local regions and compares global efficiencies for a habitat search within its local conditions based on memorized information. Here, memorized information was acquired by scanning locally perceived environments rather than remembering resource locations. When memorized information matched to its current environments, the agent changed resource selection rules. As a result, the agent revisited previous resource sites while exploring new sites, which was demonstrating a weak home-range property.